nLab
3-groupoid

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The notion of 3-groupoid is the next higher generalization in higher category theory of groupoid and 2-groupoid.

Definition

A 3-groupoid is an ∞-groupoid such that all parallel pairs of k-morphism are equivalent for k4k \geq 4: a 3-truncated ∞-groupoid.

Thus, up to equivalence, there is no point in mentioning anything beyond 33-morphisms, except whether two given parallel 33-morphisms are equivalent. This definition may give a concept more general than your preferred definition of 33-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of 33-morphisms as equality.

See also n-groupoid.

Models

A general 3-groupoid is geometrically modeled by a 4-coskeletal Kan complex. Equivalently – via the homotopy hypothesis-theorem – by a homotopy 3-type.

A small model of this is a 3-hypergroupoid, where all horn-filelrs in dimension 4\geq 4 are unique .

A 3-groupoid is algebraically modeled by a tricategory in which all morphisms are invertible, and by a 3-truncated algebraic Kan complex.

A semistrict algebraic model for 3-groupoids is provided by the notion of Gray-groupoid. These in turn are encoded by 2-crossed modules.

An entirely strict algebraic model for 3-groupoids (which no longer models all homotopy 3-types) is a 3-truncated strict omega-groupoid.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

References

  • Simona Paoli, Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids, Journal of Pure and Applied Algebra 211 (2007), 801-820. (arXiv)

  • Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv)

Revised on September 10, 2012 20:26:30 by Urs Schreiber (131.174.188.17)